Electrochemical Cells and Cell Voltage

What Is an Electrochemical Cell?

An electrochemical cell is an arrangement of two electrodes and an electrolyte (or two half-cells with a salt bridge) that allows a redox reaction to proceed with separated oxidation and reduction, so that electrical energy can be taken out (galvanic cell) or put in (electrolytic cell).

From the parent sections you already know what electrodes and electrolytes are, and that redox reactions and charge transport are involved. Here we focus on:

• How cells are built from half-reactions
• How cell voltage arises
• How to write cell notation
• Factors that influence the cell voltage

Structure of an Electrochemical Cell

Half-Cells

A half-cell consists of:

• An electrode (metal or other electronically conducting phase)
• An electrolyte solution containing ions that participate in the electrode reaction

Each half-cell hosts one half-reaction:

• Anodic half-cell: oxidation
• Cathodic half-cell: reduction

In a typical metal/metal-ion half-cell:

• Electrode: a metal (e.g. Zn)
• Electrolyte: aqueous solution of the corresponding metal ions (e.g. $\text{Zn}^{2+}$)

Example half-reactions (symbolic):

• Oxidation (anode):
  $$\text{M}(s) \rightarrow \text{M}^{n+}(aq) + n e^-$$
• Reduction (cathode):
  $$\text{M}^{n+}(aq) + n e^- \rightarrow \text{M}(s)$$

Connecting Two Half-Cells

To build a complete cell:

1. External circuit
   The two electrodes are connected by a wire. Electrons flow through this wire.
2. Ionic connection between electrolytes
   Ions must also move between the half-cells to maintain charge balance. Common arrangements:
• Salt bridge: a U-shaped tube filled with an inert electrolyte gel (e.g. KCl, KNO$_3$)
• Porous diaphragm / membrane: separates solutions but allows ions to pass

The salt bridge (or diaphragm) prevents the two solutions from mixing freely, but completes the internal ionic circuit by allowing counter-ions to move.

Direction of Charge Flow

In a galvanic cell (spontaneous reaction, electricity produced):

• Electrons flow through the external wire from anode to cathode.
• Anions typically move in the salt bridge toward the anode.
• Cations typically move in the salt bridge toward the cathode.

By convention:

• The anode is the electrode where oxidation occurs.
• The cathode is the electrode where reduction occurs.

In galvanic cells:

• Anode = negative electrode
• Cathode = positive electrode

(In electrolytic cells, the same anode/cathode definitions by reaction type apply, but the polarity is reversed by the external power source.)

Cell Reactions from Half-Reactions

An electrochemical cell is fully described by the pair of half-reactions that occur at anode and cathode.

Combining Half-Reactions

1. Write the two half-reactions one as oxidation, one as reduction.
2. Multiply by suitable integers so that the number of electrons lost = electrons gained.
3. Add the half-reactions and cancel electrons and other species that appear on both sides.

Example (abstract form):

• Anode (oxidation):
  $$\text{A} \rightarrow \text{A}^{2+} + 2 e^-$$
• Cathode (reduction):
  $$\text{B}^{+} + e^- \rightarrow \text{B}$$

Balance electrons:

• Multiply cathode half-reaction by 2:
  $\text{B}^{+} + 2e^- \rightarrow 2\text{B}$$

Add:

$$
\text{A} + 2\text{B}^{+} \rightarrow \text{A}^{2+} + 2\text{B}
$$

The resulting equation is the overall cell reaction.

The cell voltage (cell potential) corresponds to this overall reaction under given conditions.

Cell Notation (Cell Diagram)

Chemists use a compact cell notation to represent the structure of a galvanic cell without drawing a full diagram.

General form:

$$
\text{anode} \mid \text{anode-solution} \; \| \; \text{cathode-solution} \mid \text{cathode}
$$

Conventions:

• A single vertical line | indicates a phase boundary (e.g. solid | solution).
• A double vertical line || indicates a salt bridge or other ionic junction between the electrolytes.
• Species within the same phase are separated by commas.
• The anode half-cell is written on the left, the cathode half-cell on the right.

Example structure:

• Anode half-cell: $\text{M}(s) \mid \text{M}^{n+}(aq, c_1)$
• Cathode half-cell: $\text{N}^{m+}(aq, c_2) \mid \text{N}(s)$

Cell notation:

$$
\text{M}(s) \mid \text{M}^{n+}(aq, c_1) \; \| \; \text{N}^{m+}(aq, c_2) \mid \text{N}(s)
$$

If other dissolved species (e.g. supporting electrolytes, complexing agents, gases) are important, they are included in the solution part, often with concentrations or pressures.

Cell Voltage: Basic Ideas

Definition

The cell voltage (cell potential) $E_\text{cell}$ is the electrical potential difference between the two electrodes, measured under operating conditions.

In a galvanic cell:

• $E_\text{cell} > 0$ for a spontaneous overall reaction (as written).
• The maximum useful electrical work (per mole of reaction, at constant temperature and pressure, and reversible conditions) is related to $E_\text{cell}$.

We will not go into the thermodynamic derivation here; that is treated in the chapter on Gibbs free energy. Here we focus on the electrochemical viewpoint.

Origin of the Potential Difference

Each half-reaction tends to proceed in a certain direction, depending on:

• The chemical nature of the redox couple (e.g. Zn$^{2+}$/Zn, Cu$^{2+}$/Cu)
• Concentrations (activities) of the species
• Temperature, pressure

At the interface metal | solution, a charge separation forms until equilibrium is reached between electrons in the electrode and ions in solution. This gives rise to an electrode potential.

The cell voltage results from the difference between the electrode potentials of the two half-cells:

$$
E_\text{cell} = E_\text{cathode} - E_\text{anode}
$$

This expression defines $E_\text{cell}$ for a specific choice of anode and cathode (as written in cell notation).

Standard Cell Voltage and Electrode Potentials

Standard Electrode Potentials (Conceptual Use Here)

Standard electrode potentials $E^\circ$ (relative to the standard hydrogen electrode) are tabulated in the separate chapter on standard redox potentials. Here we simply use them:

• Each redox couple has a standard reduction potential $E^\circ(\text{Ox}/\text{Red})$, defined for:
• Solutes at concentration $c^\circ = 1\ \text{mol L}^{-1}$
• Gases at pressure $p^\circ = 1\ \text{bar}$
• Pure solids and liquids in their standard state
• A specified temperature (usually $25^\circ\text{C}$)

Larger $E^\circ$ (more positive) means a stronger tendency to be reduced.

Standard Cell Voltage

If both half-cells are under standard conditions, the standard cell voltage $E^\circ_\text{cell}$ is:

$$
E^\circ_\text{cell} = E^\circ_\text{cathode} - E^\circ_\text{anode}
$$

Important points:

• The $E^\circ$ values used must be reduction potentials, even for the anode. For the anode half-reaction (which actually runs as oxidation), we still insert the tabulated reduction potential and subtract it.
• $E^\circ_\text{cell}$ does not depend on how you multiply the half-reactions to balance electrons. Potentials are intensive quantities; they are not multiplied by stoichiometric coefficients.

The sign of $E^\circ_\text{cell}$ indicates the direction in which the overall standard reaction is spontaneous (positive value: spontaneous as written, negative: non-spontaneous as written).

Dependence of Cell Voltage on Concentrations: Nernst Perspective

The detailed general derivation of the Nernst equation belongs to the chapter on redox equilibria. Here we only state how it applies to entire cells and how concentration changes qualitatively affect $E_\text{cell}$.

Qualitative Effect of Concentration

For a general cell reaction:

$$
\text{aA} + \text{bB} \rightleftharpoons \text{cC} + \text{dD}
$$

with transferred electrons $z$, the reaction quotient $Q$ is (simplified, for solutes):

$$
Q = \frac{[\text{C}]^{c}[\text{D}]^{d}}{[\text{A}]^{a}[\text{B}]^{b}}
$$

The cell voltage under non-standard conditions $E_\text{cell}$ is related to $Q$ such that:

• If the products are relatively more concentrated (large $Q$), $E_\text{cell}$ is smaller than $E^\circ_\text{cell}$.
• If the reactants are relatively more concentrated (small $Q$), $E_\text{cell}$ is larger than $E^\circ_\text{cell}$.

In other words:

• Driving the system toward products (by increasing product concentrations or decreasing reactants) reduces the cell’s ability to do electrical work; $E_\text{cell}$ decreases.
• Providing more reactants increases the driving force; $E_\text{cell}$ increases, up to the limit $E^\circ_\text{cell}$ at standard conditions.

A “concentration cell” is an important special case (explained below) where $E_\text{cell}$ arises only from a difference in concentration.

Types of Electrochemical Cells (from the Viewpoint of Cell Voltage)

Galvanic vs. Electrolytic Cells

• Galvanic cell (voltaic cell):
• The overall redox reaction is spontaneous.
• $E_\text{cell} > 0$ (when written in the spontaneous direction).
• Electrical energy is produced; electrons flow from anode to cathode through the external circuit.
• Electrolytic cell:
• An external power source forces a non-spontaneous redox reaction.
• The applied voltage must exceed the reverse cell potential plus additional overpotentials.
• In operation, $E_\text{cell}$ of the cell itself is negative (in the direction of the imposed reaction); electrical energy is consumed.

The definitions of anode and cathode (by oxidation vs. reduction) remain the same in both cases, but their signs (positive/negative) differ.

Primary, Secondary, and Fuel Cells (Introductory View)

The practical behavior of $E_\text{cell}$ over time differs between cell types:

• Primary cells (batteries)
• Not (or poorly) rechargeable.
• Chemical composition changes irreversibly during discharge.
• $E_\text{cell}$ gradually decreases as reactants are consumed and concentration conditions shift.
• Secondary cells (rechargeable batteries)
• Redox reactions are (largely) reversible.
• Applying an external voltage can restore the initial composition.
• $E_\text{cell}$ rises during charging, falls during discharging, within a characteristic voltage window.
• Fuel cells
• Reactants (e.g. H$_2$, O$_2$) are continuously supplied; products are removed.
• Under appropriate flow conditions, concentrations are kept near constant, and $E_\text{cell}$ is nearly constant during operation.

Special Case: Concentration Cells

A concentration cell is an electrochemical cell where both electrodes are made of the same material and the same redox couple is involved, but the concentrations (activities) of the ionic species differ in the two half-cells.

Example structure (same metal M, different ion concentrations):

$$
\text{M}(s) \mid \text{M}^{n+}(aq, c_\text{low}) \; \| \; \text{M}^{n+}(aq, c_\text{high}) \mid \text{M}(s)
$$

• There is no difference in standard electrode potential between the two sides, because the same redox couple is involved.
• However, a potential difference (and thus $E_\text{cell} \neq 0$) arises purely from the difference in ion concentration.
• The cell reaction effectively moves ions from higher concentration to lower concentration, and electrons move in the circuit in such a way that chemical potential differences are equalized.

Key idea:

• Even when materials are identical, non-uniform conditions (different concentrations) can cause a non-zero cell voltage.

Factors Influencing Cell Voltage in Practice

While the basic theory gives $E_\text{cell}$ under ideal, reversible conditions, real cells show additional influences. Only an overview is given here; more detailed electrochemical kinetics is treated elsewhere.

1. Concentration and Activity

As mentioned, ion concentrations (really, activities) affect electrode potentials, and hence $E_\text{cell}$. Typical practical consequences:

• As a battery discharges, reactant concentrations decrease and product concentrations increase; $E_\text{cell}$ drops.
• Highly concentrated or mixed electrolytes can deviate from ideal behavior, so activity coefficients become important.

2. Temperature

Raising temperature generally changes both:

• Equilibrium constants of the redox reactions
• The relationship between potential and concentration

Thus, $E_\text{cell}$ is temperature-dependent. Some cells are designed for a specific operating temperature range to ensure a stable voltage.

3. Gas Pressure (for Gas-Electrode Cells)

In cells involving gases (e.g. O$_2$/H$_2$ fuel cells, Cl$_2$ electrodes):

• The gas partial pressures influence the electrode potentials.
• At higher gas pressure, the gas’s effective concentration (activity) is higher, usually leading to higher $E_\text{cell}$ for reactions that consume that gas.

4. Internal Resistance and Overpotentials

Ideal theory assumes that the cell is reversible and that no energy is lost to resistive or kinetic effects. Real cells, however, have:

• Internal resistance: resistance of electrolytes, electrodes, separators, and contacts.
• When a current flows, some voltage is dropped inside the cell; the measured terminal voltage under load is lower than the open-circuit $E_\text{cell}$.
• Overpotentials (overvoltages): additional voltage required to drive electrode reactions at a finite rate.
• Activation overpotential (kinetics of electron transfer)
• Concentration overpotential (mass transport limitations)
• Gas evolution overpotential (e.g. for H$_2$, O$_2$ evolution)

These effects do not change the equilibrium $E_\text{cell}$ but affect the operating voltage under load, especially at higher current densities.

Measuring and Using Cell Voltage

Open-Circuit Voltage vs. Loaded Voltage

• Open-circuit voltage (OCV):
  The voltage measured when no current flows (the cell is connected to a high-impedance voltmeter only).
• This is the closest practical approximation to the theoretical $E_\text{cell}$.
• Loaded voltage:
  The voltage when the cell is delivering current to an external circuit.
• Due to internal resistance and overpotentials, this is lower than the OCV.

The relationship between current, internal resistance $R_\text{int}$, and voltage drop can be approximated (for small currents) using Ohm’s law:

$$
\Delta V_\text{int} \approx I \cdot R_\text{int}
$$

where $\Delta V_\text{int}$ is the internal voltage loss, and $I$ is the current.

Polarity and Connection of Cells

From $E_\text{cell}$ we know which electrode is positive or negative in galvanic mode. When building battery packs:

• Series connection of cells:
• The voltages add.
• Total voltage $E_\text{total} \approx n \times E_\text{cell}$ (for $n$ identical cells).
• Parallel connection:
• The voltage remains approximately the same.
• The available current capacity increases.

Correct orientation (matching plus/minus poles) is crucial; reversing a cell in a string can lead to charging it in the wrong direction, heating, or damage.

Summary

• An electrochemical cell consists of two half-cells, each with an electrode and an electrolyte, connected by both an external electronic path and an internal ionic path (salt bridge or membrane).
• The overall cell reaction is obtained by combining the oxidation and reduction half-reactions such that electrons cancel.
• Cell notation gives a compact description of the cell: anode on the left, cathode on the right, with | for phase boundaries and || for the salt bridge.
• The cell voltage $E_\text{cell}$ is the difference between the electrode potentials of cathode and anode: $E_\text{cell} = E_\text{cathode} - E_\text{anode}$.
• Under standard conditions, the cell voltage is $E^\circ_\text{cell}$, obtained from tabulated standard electrode (reduction) potentials.
• Non-standard concentrations, temperatures, and pressures change $E_\text{cell}$; the qualitative dependence is described via the reaction quotient.
• Galvanic cells deliver electrical energy (spontaneous reaction, $E_\text{cell} > 0$); electrolytic cells consume electrical energy to drive non-spontaneous reactions.
• Special cases like concentration cells show that even identical materials can produce a voltage if the environments differ.
• Real cell voltages under load are lower than the ideal $E_\text{cell}$ because of internal resistance and overpotentials, important for practical applications such as batteries and fuel cells.